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EE3054 Signals and Systems
Continuous Time Convolution
Yao WangPolytechnic University
Some slides included are extracted from lecture presentations prepared by McClellan and Schafer
3/14/2008 © 2003, JH McClellan & RW Schafer 2
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3/14/2008 © 2003, JH McClellan & RW Schafer 3
LECTURE OBJECTIVES
� Review of C-T LTI systems � Evaluating convolutions
� Examples� Impulses
� LTI Systems� Stability and causality� Cascade and parallel connections
3/14/2008 © 2003, JH McClellan & RW Schafer 4
Linear and Time-Invariant (LTI) Systems
� If a continuous-time system is both linear and time-invariant, then the output y(t) is related to the input x(t) by a convolution integralconvolution integral
where h(t) is the impulse responseimpulse response of the system.
∫∞
∞−
∗=−= )()()()()( thtxdthxty τττ
3/14/2008 © 2003, JH McClellan & RW Schafer 5
Evaluating a Convolution
∫∞
∞−
∗=−= )()()()()( txthdtxhty τττ
)()( tueth t−=)1()( −= tutx
3/14/2008 © 2003, JH McClellan & RW Schafer 6
“Flipping and Shifting”)(τx
g(τ ) = x(−τ ) = u(−τ −1)
g(τ − t) = x(−(τ − t)) = x(t −τ )
“flipping”
“flipping and shifting”
t1−t
3/14/2008 © 2003, JH McClellan & RW Schafer 7
Evaluating the Integral
>−
<−=
∫−
− 01
010)( 1
0
tde
tty t
ττ
3/14/2008 © 2003, JH McClellan & RW Schafer 8
Solution
1 0)( t<ty =
1 1
)(
)1(
1
0
1
0
≥−=
−==
−−
− −−−∫
te
edety
t
t tττ τ
3/14/2008 © 2003, JH McClellan & RW Schafer 9
Convolution GUI
3/14/2008 © 2003, JH McClellan & RW Schafer 10
Another Example
)(00
0
00
0)()(
)()()()()(
0)(
tuabee
t
tba
ee
t
tdeeedtueue
thtxdthxty
btatbtat
tbabt
tba
−−=
<
>+−
−=
<
>=−=
∗=−=
−−−−
−−∞
∞−
−−−
∞
∞−
∫∫
∫
ττττ
τττ
ττττ
)()( tuetx at−= abtueth bt ≠= − ),()(
3/14/2008 © 2003, JH McClellan & RW Schafer 11
Special Case: u(t)
)()1(1
)()()()()(
tuea
thtxdthxty
at−
∞
∞−
−=
∗=−= ∫ τττ
0),()( ≠= − atuetx at )()( tuth =
)()1(21)(
2 if2 tuety
at−−=
=
3/14/2008 © 2003, JH McClellan & RW Schafer 12
Convolve Unit Steps
)(00
000
01)()(
)()()()()(
0
tutt
ttt
tddtuu
thtxdthxty
t
=
<
>=
<
>=−=
∗=−=
∫∫
∫
∞
∞−
∞
∞−
ττττ
τττ
)()( tutx = )()( tuth =
Unit Ramp
3/14/2008 © 2003, JH McClellan & RW Schafer 13
“Flipping and Shifting”x(τ )
g(τ ) = x(−τ ) = u(−τ −1)
g(τ − t) = x(−(τ − t)) = x(t −τ )
“flipping”
“flipping and shifting”
t1−t
More examples
� Rectangular pulses
3/14/2008 © 2003, JH McClellan & RW Schafer 15
Another Convolution Example
y(t) = x(τ )h(t −τ )dτ = x(t) ∗ h(t)−∞
∞∫
h(t) = e− tu(t)
3/14/2008 © 2003, JH McClellan & RW Schafer 16
Evaluating the Integral
y(t) = 0 t <1
= e−(t−τ )dτ 1≤ t ≤ 21
t∫
= e−(t−τ )dτ 2 ≤ t1
2∫
3/14/2008 © 2003, JH McClellan & RW Schafer 17
Solution
y(t) = e−(t−τ )dτ = e−(t−τ )1
t=1-e−(t−1) 1≤ t ≤ 2
1
t∫
= e−(t−τ )dτ = e−(t−τ )1
2= e−(t−2) - e−(t−1) 2 ≤ t
1
2∫
y(t) = 0 t <1
3/14/2008 © 2003, JH McClellan & RW Schafer 18
Convolution GUI
3/14/2008 © 2003, JH McClellan & RW Schafer 19
Convolution with Impulses, etc.
� Convolution with impulses
� Convolution with step function = integrator
)()(*)( 11 ttxtttx −=−δ
3/14/2008 © 2003, JH McClellan & RW Schafer 20
Convolution is Commutative
∫
∫
∫
∞
∞−
∞−
∞
∞
∞−
∗=−=
−−=∗
−=−=
−=∗
)()()()(
)()()()(
and let
)()()()(
thtxdxth
dxthtxth
ddt
dtxhtxth
σσσ
σσσ
τστσ
τττ
3/14/2008 © 2003, JH McClellan & RW Schafer 21
Stability
� A system is stable if every bounded input produces a bounded output.
� A continuous-time LTI system is stable if and only if
h(t)dt < ∞−∞
∞∫
Integrator is unstable
3/14/2008 © 2003, JH McClellan & RW Schafer 23
Causal Systems
� A system is causal if and only if y(t0)depends only on x(τ) for τ< t0 .
� An LTI system is causal if and only if
0for 0)( <= tth
3/14/2008 © 2003, JH McClellan & RW Schafer 24
� Substitute x(t)=ax1(t)+bx2(t)
Therefore, convolution is linear.
)()(
)()()()(
)()]()([)(
21
21
21
tbytay
dthxbdthxa
dthbxaxty
+=
−+−=
−+=
∫ ∫
∫∞
∞−
∞
∞−
∞
∞−
ττττττ
ττττ
Convolution is Linear
3/14/2008 © 2003, JH McClellan & RW Schafer 25
Convolution is Time-Invariant
� Substitute x(t-t0)
w(t) = h(τ )x((t − τ ) −to)dτ−∞
∞
∫
= h(τ )x((t −to ) −τ )dτ−∞
∞
∫
= y(t − to )
3/14/2008 © 2003, JH McClellan & RW Schafer 26
Cascade of LTI Systems
δ(t) h1(t) h1(t) ∗ h2(t)
δ(t) h2 (t) h2 (t) ∗ h1(t)
h(t) = h1(t) ∗ h2 (t) = h2(t) ∗ h1(t)
3/14/2008 © 2003, JH McClellan & RW Schafer 27
Parallel LTI Systems
h(t) = h1(t) + h2(t)(t
δ(t)
h1(t)
h2 (t)
h1(t) + h2 (t)
Example: More complicated combinations
READING ASSIGNMENTS
� This Lecture:� Chapter 9, Sects. 9-6, 9-7, and 9-8
� Other Reading:� Ch. 9, all� Next Lecture: Start reading Chapter 10
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